class: center, middle, inverse, title-slide # IDS 702: Module 6.1 ## The potential outcomes framework and causal estimands ### Dr. Olanrewaju Michael Akande --- ## Causality --- <i class="fa fa-quote-left fa-2x fa-pull-left fa-border" aria-hidden="true"></i> <i class="fa fa-quote-right fa-2x fa-pull-right fa-border" aria-hidden="true"></i> We do not have knowledge of a thing until we have grasped its why, that is to say, <font color="red">its cause</font>. -- Aristotle, Physics -- </br> - Over the next few modules, we will discuss causal inference, specifically, on measuring the .hlight[effects of causes]. -- - For now, we will simply lay the foundations for causal inference. -- - We will get more into the actual methods later. --- ## Association vs. causation - In the models we have covered so far, our focus has been on inferring .hlight[associations] using samples drawn from our population of interest. -- - For example, we have been asking questions such as, do people who receive job training tend to earn more wages than people who do not? -- - Causal inference goes further as we try to infer aspects of the actual data generating process, that is, .hlight[causation]. -- - For example, does receiving job training actually cause one to earn more wage than they would have without the training? -- - The additional information needed to move from association to causation is often provided by .hlight[causal assumptions] (often untestable). -- - Note: in most cases, .hlight[association does not imply causation]! --- ## Confounding - Why is it that association does not often imply causation? .hlight[confounding variables or confounders]! -- - Causal relationship <div id="htmlwidget-cfff5415593048e9fc33" style="width:2100px;height:180px;" class="DiagrammeR html-widget"></div> <script type="application/json" data-for="htmlwidget-cfff5415593048e9fc33">{"x":{"diagram":"\n\t graph LR\n\t W(Treatment)-->Y(Outcome)\n\t "},"evals":[],"jsHooks":[]}</script> -- - Confounding <div id="htmlwidget-6a2ea320d8096f150c3c" style="width:2100px;height:180px;" class="DiagrammeR html-widget"></div> <script type="application/json" data-for="htmlwidget-6a2ea320d8096f150c3c">{"x":{"diagram":"\n\t graph TB\n\t C(Confounder)-->W(Treatment)\n\t C-->Y(Outcome)\n\t "},"evals":[],"jsHooks":[]}</script> --- ## Examples of confounding - Ice cream consumption and number of people who drowned. .hlight[Confounder: temperature]; people tend to consume more ice cream and also swim more when it is hot. -- - Medical treatment and patient outcome. .hlight[Confounders: age, sex, other complications] -- - Education and income. .hlight[Confounder: socio-economic status of family] -- - An extreme example of confounding is Simpson’s paradox: where a confounder reverses the sign of the correlation between treatment and outcome --- ## Simpson's paradox - Example: kidney stone treatment (Charig et al., BMJ, 1986). + Compare the success rates of two treatments for kidney stones -- + Treatment A: open surgery. Treatment B: small puncture -- <br /> | Treatment A | Treatment B | :----------- | :--------- | :--------- | Small stones | <b>93%</b> (81/87) | 87% (234/270) | Large stones | <b>73%</b> (192/263) | 69% (55/80) | Both | 78% (273/350) | <b>83%</b> (289/350) | -- + Overall treatmnent B has a higher success rate but treatment A actually has higher success rates given the type of stones. -- + What is the confounder here? Severity of the case/type of stones. --- ## Simpson's paradox or Yule-Simpson effect - Simpson’s paradox: a trend appears in different groups of data but disappears or reverses when these groups are combined. <img src="img/SimpsonsParadox.png" width="400px" height="300px" style="display: block; margin: auto;" /> -- - Mathematically, it is about conditioning. -- - Another well-known example is the Berkeley admission gender bias (Bickel et al., Science, 1976). --- ## General notation - .hlight[W]: Treatment (e.g. job training); we will focus on binary treatments. - .hlight[Y]: Outcome (e.g. annual wages). - .hlight[X]: Observed predictors or confounders (e.g. age, education, etc). - .hlight[U]: Unobserved predictors or confounders. </br> -- - Examples of causal questions: + Causal effect of exposure to a disease. + Comparative effectiveness research such as in clinical trials: whether one drug or medical procedure is better than the other. + Program evaluation in economics and policy. --- class: center, middle # Potential outcomes framework --- ## Potential outcomes framework - The .hlight[potential outcomes framework] or .hlight[counterfactual framework] or .hlight[Rubin Causal Model (RCM)] is arguably the most widely used framework across many disciplines, e.g., medicine, health care, policy, social sciences. -- - Under this framework, causal inference is viewed as a problem of missing data with explicit mathematical modeling of the assignment mechanism as a process for revealing the observed data. -- - Rooted in the statistical work on randomized experiments by Fisher (1918, 1925) and Neyman (1923), as extended by Rubin (1974, 1976, 1977, 1978, 1990). --- ## Potential outcomes framework - For a binary treatment, each individual gets exactly one of the two options, and we observe the corresponding response for that. -- - Conceptually, under the potential outcomes framework, we think about what each individual's response should have been had they gotten the other treatment option instead of the one they actually got. -- - The individual causal effect then is the difference between the two "potential" outcomes, only one of which is observed. -- - Clearly, we never observe the two potential outcomes for any individual, making it natural to think of this as a missing data problem. --- ## Potential outcomes framework - No causation without manipulation - "cause" must be (hypothetically speaking) something we can manipulate. e.g., intervention, action, treatment. -- - That is, gender, time and age are not well defined “causes" under the RCM. -- - Three integral components of the potential outcomes framework: + .hlight[potential outcomes] corresponding to the various levels of a treatment. + .hlight[assignment mechanisms], that is, the treatment indicator for all observations. + a .hlight[model] for the science (the potential outcomes and covariates). --- ## Potential outcomes framework: basic concepts - .hlight[Unit]: The person, place, or thing upon which a treatment will operate, at a particular time (note: a single person, place, or thing at two different times comprises two different units). -- - .hlight[Treatment]: An intervention, the effects of which (on some particular measurement of the units) the investigator wishes to assess relative to no intervention (i.e., the control). -- - .hlight[Potential Outcomes]: The values of a unit’s measurement of interest after (a) application of the treatment and (b) non-application of the treatment (i.e., under control). -- - .hlight[Causal Effect]: For each unit, the comparison of the potential outcome under treatment and the potential outcome under control. --- ## Causal effects - For a single unit, let `\(Y(0)\)` denote the outcome given the control treatment and `\(Y(1)\)`, the outcome given the active treatment. -- - For example, suppose `\(Y\)` denotes a score (level of severity) for headache, then for a single unit, we could have -- <table> <caption>Raw scores</caption> <tr> <th>Unit</th> <th>Initial headache</th> <th height="30px" colspan="2">Potential outcomes</th> <th>Causal effect</th> </tr> <tr> <th> </th> <td height="30px" style="text-align:center" width="100px"> X </td> <td style="text-align:center" width="100px"> Y(asp) </td> <td style="text-align:center" width="100px"> Y(not) </td> <td style="text-align:center" width="250px"> Y(asp) - Y(not) </td> </tr> <tr> <td height="30px" style="text-align:center" width="80px"> you</td> <td style="text-align:center"> 80 </td> <td style="text-align:center"> 25 </td> <td style="text-align:center"> 75 </td> <td style="text-align:center"> -50 </td> </tr> </table> -- <table> <caption>Gain scores</caption> <tr> <th>Unit</th> <th>Initial headache</th> <th height="30px" colspan="2">Potential outcomes</th> <th>Causal effect</th> </tr> <tr> <th> </th> <td height="30px" style="text-align:center" width="100px"> X </td> <td style="text-align:center" width="100px"> Y(asp) - X </td> <td style="text-align:center" width="100px"> Y(not) - X </td> <td style="text-align:center" width="250px"> [Y(asp) - X] - [Y(not) - X] </td> </tr> <tr> <td height="30px" style="text-align:center" width="80px"> you</td> <td style="text-align:center"> 80 </td> <td style="text-align:center"> -55 </td> <td style="text-align:center"> -5 </td> <td style="text-align:center"> -50 </td> </tr> </table> --- ## The fundamental problem of causal inference As mentioned before, - The fundamental problem of causal inference: we can observe at most one of the potential outcomes for each unit. -- - Causal inference under the potential outcome framework is essentially a missing data problem. -- - To identify causal effects from observed data, under any mathematical framework, one must make assumptions (structural or/and stochastic) -- - Since we can at most observe a single potential outcome, we must rely on multiple units (and a lot of assumptions) to make causal inferences. --- ## Basic setup - .hlight[Target population]: a well-defined population of individuals whose outcomes are going to be compared -- - .hlight[Data]: a random sample of `\(N\)` units from a target population. -- - A treatment with two levels: `\(w = 0,1\)`. -- - For each unit `\(i\)`, we observe + the binary treatment status `\(W_i \in \{0,1\}\)`, + a vector of `\(p\)` predictors/covariates `\(X_i = (X_{i1}, \ldots, X_{ip})\)`, and + an outcome `\(Y_i^{\text{obs}}\)`. --- ## Basic setup - For each unit `\(i\)`, there are two potential outcomes `\((Y_i(0),Y_i(1))\)`. -- - That is, the outcomes under the two values of the treatment, at most one of which is observed. -- - Potential outcomes and assignments jointly determine the values of the observed outcomes .block[ $$ Y_i^{\text{obs}} \equiv Y_i(W_i) = W_i \cdot Y_i(1) + (1-W_i) \cdot Y_i(0) $$ ] -- and the missing outcomes: .block[ $$ Y_i^{\text{mis}} \equiv Y_i(1-W_i) = (1-W_i) \cdot Y_i(1) + W_i \cdot Y_i(0) $$ ] --- ## Causal estimands - The .hlight[average treatment effect (ATE)]: .block[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0)]. $$ ] -- - The .hlight[average treatment effect for the treated (ATT)]: .block[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0) | W_i = 1]. $$ ] -- - The .hlight[average treatment effect for the control (ATC)]: .block[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0) | W_i = 0]. $$ ] -- - For binary outcomes, .hlight[causal odds ratio (OR) or risk ratio (RR):]: .block[ $$ \tau = \dfrac{\mathbb{Pr}[Y_i(1) = 1]/\mathbb{Pr}[Y_i(1) = 0]}{\mathbb{Pr}[Y_i(0) = 1]/\mathbb{Pr}[Y_i(0) = 0]}. $$ ] - Obviously these estimands are not identifiable without further assumptions. -- - We will start to explore those soon. --- ## Example <table> <tr> <th> </th> <th height="30px" colspan="2">Potential Outcomes</th> <th> </th> <th height="30px" colspan="3">Observed Data</th> </tr> <tr> <th> </th> <td height="30px" style="text-align:center" width="100px"> Y(0) </td> <td style="text-align:center" width="100px"> Y(1) </td> <td style="text-align:center" width="100px"> </td> <td style="text-align:center" width="100px"> W </td> <td height="30px" style="text-align:center" width="100px"> Y(0) </td> <td style="text-align:center" width="100px"> Y(1) </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 13 </td> <td style="text-align:center"> 14 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 1 </td> <td style="text-align:center"> ? </td> <td style="text-align:center"> 14 </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> ? </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 4 </td> <td style="text-align:center"> 1 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> 4 </td> <td style="text-align:center"> ? </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 5 </td> <td style="text-align:center"> 2 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> 5 </td> <td style="text-align:center"> ? </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> 3 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> ? </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> 1 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 0 </td> <td style="text-align:center"> 6 </td> <td style="text-align:center"> ? </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 8 </td> <td style="text-align:center"> 10 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 1 </td> <td style="text-align:center"> ? </td> <td style="text-align:center"> 10 </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> </td> <td style="text-align:center"> 8 </td> <td style="text-align:center"> 9 </td> <td style="text-align:center"> </td> <td style="text-align:center"> 1 </td> <td style="text-align:center"> ? </td> <td style="text-align:center"> 9 </td> </tr> <tr> <td height="30px" style="text-align:center" width="100px"> True averages </td> <td style="text-align:center"> 7 </td> <td style="text-align:center"> 5 </td> <td style="text-align:center"> Observed averages </td> <td style="text-align:center"> </td> <td style="text-align:center"> 5.4 </td> <td style="text-align:center"> 11 </td> </tr> </table> --- ## Acknowledgements These slides contain materials adapted from courses taught by Dr. Fan Li. --- class: center, middle # What's next? ### Move on to the readings for the next module!